11 REGULARIZATION 301 (b) With the help of Problem 10, construct a sequence (w() of continuous positive functions, with supports contained in V2, and such that || wt\\ ^ 1 for all /, the sequence (wt) being uniformly bounded and equicontinuous and such that the set of functions w,(YO*>'i wj) is uniformly bounded as zt runs through Ut for each / ^> 1 . (c) With the help of Problem 9, construct a sequence (gf) of positive functions with supports contained in V4, which is uniformly bounded and equicontinuous and such that the set of functions rti(^(zi)gi gt) is uniformly bounded and equicontinuous as zi runs through U, for each / 2> 1 ; and such that for each function / which is the uniform limit of a subsequence of the sequence (gt), we have f(x)<f(e) whenever 12. Let G be a locally compact group with no small subgroups, and let V be a symmetric compact neighborhood of>, containing no subgroup of G other than {e}, and such that for x, y e V the relation x2 = y2 implies that x = y (Section 12.9, Problem 6(a)). A continuous homomorphism rh~* X(r) of R into G is called a one-parameter subgroup of G (Section 12.9, Problem 7). The constant one-parameter subgroup r\ *e is de- noted by 0. For each / e R, the one-parameter subgroup r\ *X(tr) is denoted by tX. If X is a one-parameter subgroup of G, a continuous real-valued function / on G is said to be X-differentiable if the function r~l(^((X(r))ff) converges uniformly on G to a limit D*/ as r-^-0 in R. If so, then / is also /JSf-differentiable for all real t, and we have T>txf= tDxf. If /is Jf-differentiable, then for each x e G the function r\-+f(X(r)x) is differentiate on R, and its derivative is the function A continuous bounded real-valued function /on G is said to be adequate if there exists a neighborhood V of e in G such that || Y(z)f f\\ =£ 0 for all z e V. The functions /constructed in Problem ll(c) as limits of subsequences of the sequence (gf) are adequate. Show that, if /is adequate and if D*/ exists for some one-parameter sub- group X T£ 0, then Dxf^ 0. If on the contrary/is not adequate, then there exists a one- parameter subgroup X ^ 0 such that D*/= 0. (Consider a sequence (a,) of elements ^e in G, tending to e, and such that ||Y(a/)/ /ll = 0- Show that it can be assumed that there exists a sequence of integers ra/ with the property that for all r e R the sequence a^mj^ converges to X(r), where X^Q (Section 12.9, Problem 7).) 13. The notation and assumptions on G and V remain the same as in Problem 12. Let (fj) be a sequence of bounded continuous functions on G which converges uniformly on G to a function /. Let (aj) be a sequence of elements of G tending to e, and (mj) a sequence of integers >0. (a) Suppose that /is adequate and that the functions mfy(aj)fj /}) are uniformly bounded. Then, for each neighborhood U of e, there exists a number e > 0 such that the relation \r\ < e implies a1^ e U. (Suppose that U2 c: V. Let by taking / sufficiently large, we may suppose that a/ e U and \\y(z)fj fj\\ ^ j8 > 0 for all z 6 V U. Show that we cannot have a] e V U for k < <xmj//3, by using the inequality (b) Suppose that the functions mj(j(aj)fj - fj) are uniformly bounded and uniformly equicontinuous, and also that for each r e R the sequence (a^mj]) tends to X(r)t where X is a one-parameter subgroup. Show that Dxf exists and is the limit of the - e for